Engineering surveying, 5...ition w. schofield

This book is dedicated to my late wife Jean and my daughter Zoë

Engineering Surveying
Theory and Examination
Problems for Students
Fifth Edition
W. Schofield
Principal Lecturer, Kingston University
OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI

Butterworth-Heinemann
Linacre House, Jordan Hill, Oxford OX2 8DP
225 Wildwood Avenue, Woburn, MA 01801-2041
A division of Reed Educational and Professional Publishing Ltd
A member of the Reed Elsevier plc group
First published 1972
Second edition 1978
Third edition 1984
Fourth edition 1993
Reprinted 1995, 1997, 1998
Fifth edition 2001
© W. Schofield 1972, 1978, 1984, 1993, 1998, 2001
All rights reserved. No part of this publication
may be reproduced in any material form (including
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to some other use of this publication) without the
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licence issued by the Copyright Licensing Agency Ltd,
90 Tottenham Court Road, London, England W1P 9HE.
Applications for the copyright holder’s written permission
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British Library Cataloguing in Publication Data
Schofield, W. (Wilfred)
Engineering surveying: theory and examination problems for students. – 5th ed.
1 Surveying
I Title
526.9′024624
Library of Congress Cataloguing in Publication Data
Schofield, W. (Wilfred)
Engineering surveying: theory and examination problems for students/W. Schofield.– 5th ed.
p. cm.
ISBN 0 7506 4987 9 (pbk.)
1 Surveying I Title.
TA545.S263 2001
526.9′024′62–dc21
ISBN 0 7506 4987 9
Typeset in Replika Press Pvt Ltd. 100% EOU, Delhi 110 040, (India)
Printed and bound in Great Britain

Contents
Preface to fifth edition vii
Preface to fourth edition ix
Acknowledgements xi
1 Basic concepts of surveying 1
Definition – Basic measurements – Control networks – Locating position – Locating
topographic detail – Computer systems – DGM – CAD – GIS – Vector/raster – Topology –
Laser scanner – Summary – Units of measurement – Significant figures – Rounding off
numbers – Errors in measurement – Indices of precision – Weight – Rejection of outliers –
Combination of errors
2 Vertical control 43
Introduction – Levelling – Definitions – Curvature and refraction – Equipment – Instrument
adjustment – Principle of levelling – Sources of error – Closure tolerances – Error
distribution – Levelling applications – Reciprocal levelling – Precise levelling – Digital
levelling – Trigonometrical levelling – Stadia tacheometry
3 Distance 117
Tapes – Field work – Distance adjustment – Errors in taping – Accuracies –
Electromagnetic distance measurement (EDM) – Measuring principles – Meteorological
corrections – Geometrical reductions – Errors and calibration – Other error sources –
Instrument specifications – Developments in EDM – Optical distance measurement (ODM)
4 Angles 178
The theodolite – Instrumental errors – Instrument adjustment – Field procedure – Measuring
angles – Sources of error
5 Position 208
Introduction – Reference ellipsoid – Coordinate systems – Local systems – Computation on
the ellipsoid – Datum transformations – Orthomorphic projection – Ordnance Survey
National Grid – Practical applications – The Universal Transverse Mercator Projection
(UTM) – Plane rectangular coordinates
6 Control surveys 252
Traversing – Triangulation – Trilateration – Triangulateration – Inertial surveying
7 Satellite positioning 307
Introduction – GPS segments – GPS receivers – Satellite orbits – Basic principle of position
fixing – Differencing data – GPS field procedures – Error sources – GPS survey planning –
Transformation between reference systems – Datums – Other satellite systems –
Applications

8 Curves 347
Circular curves – Setting out curves – Compound and reverse curves – Short and/or small-
radius curves – Transition curves – Setting-out data – Cubic spiral and cubic parabola –
Curve transitional throughout – The osculating circle – Vertical curves
9 Earthworks 420
Areas – Partition of land – Cross-sections – Dip and strike – Volumes – Mass-haul
diagrams
10 Setting out (dimensional control) 464
Protection and referencing – Basic setting-out procedures using coordinates – Technique for
setting out a direction – Use of grids – Setting out buildings – Controlling verticality – Controlling
grading excavation – Rotating lasers – Laser hazards – Route location – Underground surveying
– Gyro-theodolite – Line and level – Responsibility on site – Responsibility of the setting-out
engineer
Index 517
vi Contents

Preface to the fourth edition
This book was originally intended to combine volumes 1 and 2 of Engineering Surveying, 3rd and
2nd editions respectively. However, the technological developments since the last publication date
(1984) have been so far-reaching as to warrant the complete rewriting, modernizing and production
of an entirely new book.
Foremost among these developments are the modern total stations, including the automatic self-
seeking instruments; completely automated, ‘field to finish’ survey systems; digital levels; land/
geographic information systems (L/GIS) for the managing of any spatially based information or
activity; inertial survey systems (ISS); and three-dimensional position fixing by satellites (GPS).
In order to include all this new material and still limit the size of the book a conscious decision
was made to delete those topics, namely photogrammetry, hydrography and field astronomy, more
adequately covered by specialist texts.
In spite of the very impressive developments which render engineering surveying one of the
most technologically advanced subjects, the material is arranged to introduce the reader to elementary
procedures and instrumentation, giving a clear understanding of the basic concept of measurement
as applied to the capture, processing and presentation of spatial data. Chapters 1 and 4 deal with the
basic principles of surveying, vertical control, and linear and angular measurement, in order to
permit the student early access to the associated equipment. Chapter 5 deals with coordinate
systems and reference datums necessary for an understanding of satellite position fixing and an
appreciation of the various forms in which spatial data can be presented to an L/GIS. Chapter 6
deals with control surveys, paying particular attention to GPS, which even in its present incomplete
stage has had a revolutionary impact on all aspects of surveying. Chapter 7 deals with elementary,
least squares data processing and provides an introduction to more advanced texts on this topic.
Chapters 8 to 10 cover in detail those areas (curves, earthworks and general setting out on site) of
specific interest to the engineer and engineering surveyor. Each chapter contains a section of
‘Worked Examples’, carefully chosen to clearly illustrate the concepts involved. Student exercises,
complete with answers, are supplied for private study. The book is aimed specifically at students of
surveying, civil, mining and municipal engineering and should also prove valuable for the continuing
education of professionals in these fields.
W. Schofield

This Page Intentionally Left Blank

Preface to the fifth edition
Since the publication of the fourth edition of this book, major changes have occurred in the
following areas:
• surveying instrumentation, particularly Robotic Total Stations withAutomatic Target Recognition,
reflectorless distance measurement, etc., resulting in turnkey packages for machine guidance
and deformation monitoring. In addition there has been the development of a new instrument
and technique known as laser scanning
• GIS, making it a very prominent and important part of geomatic engineering
• satellite positioning, with major improvements to the GPS system, the continuance of the GLONASS
system, and a proposal for a European system called GALILEO
• national and international co-ordinate systems and datums as a result of the increasing use of
satellite systems.
All these changes have been dealt with in detail, the importance of satellite systems being
evidenced by a new chapter devoted entirely to this topic.
In order to include all this new material and still retain a economical size for the book, it was
necessary but regrettable to delete the chapter on Least Squares Estimation. This decision was
based on a survey by the publishers that showed this important topic was not included in the
majority of engineering courses. It can, however, still be referred to in the fourth edition or in
specialised texts, if required.
All the above new material has been fully expounded in the text, while still retaining the many
worked examples which have always been a feature of the book. It is hoped that this new edition
will still be of benefit to all students and practitioners of those branches of engineering which
contain a study and application of engineering surveying.
W. Schofield
February 2001

Acknowledgements
The author wishes to acknowledge and thank all those bodies and individuals who contributed in
any way to the formation of this book.
For much of the illustrative material thanks are due to Intergraph (UK) Ltd, Leica (UK) Ltd,
Trimble (UK) Ltd, Spectra-Precision Ltd, Sokkisha (UK) Ltd, and the Ordnance Survey of Great
Britain (OSGB).
I am also indebted to OSGB for their truly excellent papers, particularly ‘A Guide to Co-ordinate
Systems in Great Britain’, which formed the basis of much of the information in chapter 7.
I must also acknowledge the help received from the many papers, seminars, conferences, and
continued quality research produced by the IESSG of the University of Nottingham.
Finally, may I say thank you to Pat Affleck of the Faculty of Technology, Kingston University,
who freely and unstintingly typed all this new material.

1
Basic concepts of surveying
The aim of this chapter is to introduce the reader to the basic concepts of surveying. It is therefore
the most important chapter and worthy of careful study and consideration.
1.1 DEFINITION
Surveying may be defined as the science of determining the position, in three dimensions, of
natural and man-made features on or beneath the surface of the Earth. These features may then be
represented in analog form as a contoured map, plan or chart, or in digital form as a three-
dimensional mathematical model stored in the computer. This latter format is referred to as a digital
ground model (DGM).
In engineering surveying, either or both of the above formats may be utilized in the planning,
design and construction of works, both on the surface and underground. At a later stage, surveying
techniques are used in the dimensional control or setting out of the designed constructional elements
and also in the monitoring of deformation movements.
In the first instance, surveying requires management and decision making in deciding the appropriate
methods and instrumentation required to satisfactorily complete the task to the specified accuracy
and within the time limits available. This initial process can only be properly executed after very
careful and detailed reconnaissance of the area to be surveyed.
When the above logistics are complete, the field work – involving the capture and storage of field
data – is carried out using instruments and techniques appropriate to the task in hand.
The next step in the operation is that of data processing. The majority, if not all, of the computation
will be carried out by computer, ranging in size from pocket calculator to mainframe. The methods
adopted will depend upon the size and precision of the survey and the manner of its recording;
whether in a field book or a data logger. Data representation in analog or digital form may now be
carried out by conventional cartographic plotting or through a totally automated system using a
computer-driven flat-bed plotter. In engineering, the plan or DGM is used for the planning and
design of a construction project. This project may comprise a railroad, highway, dam, bridge, or
even a new town complex. No matter what the work is, or how complicated, it must be set out on
the ground in its correct place and to its correct dimensions, within the tolerances specified. To this
end, surveying procedures and instrumentation are used, of varying precision and complexity,
depending on the project in hand.
Surveying is indispensable to the engineer in the planning, design and construction of a project,
so all engineers should have a thorough understanding of the limits of accuracy possible in the
construction and manufacturing processes. This knowledge, combined with an equal understanding
of the limits and capabilities of surveying instrumentation and techniques, will enable the engineer
to successfully complete his project in the most economical manner and shortest time possible.

2 Engineering Surveying
1.2 BASIC MEASUREMENTS
Surveying is concerned with the fixing of position whether it be control points or points of topographic
detail and, as such, requires some form of reference system.
The physical surface of the Earth, on which the actual survey measurements are carried out, is
mathematically non-definable. It cannot therefore be used as a reference datum on which to compute
position.
An alternative consideration is a level surface, at all points normal to the direction of gravity.
Such a surface would be formed by the mean position of the oceans, assuming them free from all
external forces, such as tides, currents, winds, etc. This surface is called the geoid and is the
equipotential surface at mean sea level. The most significant aspect of this surface is that survey
instruments are set up relative to it. That is, their vertical axes, which are normal to the plate bubble
axes used in the setting-up process, are in the direction of the force of gravity at that point. Indeed,
the points surveyed on the physical surface of the Earth are frequently reduced to their equivalent
position on the geoid by projection along their gravity vectors. The reduced level or elevation of a
point is its height above or below the geoid as measured in the direction of its gravity vector (or
plumb line) and is most commonly referred to as its height above or below mean sea level (MSL).
However, due to variations in the mass distribution within the Earth, the geoid is also an irregular
surface which cannot be used for the mathematical location of position.
The mathematically definable shape which best fits the shape of the geoid is an ellipsoid formed
by rotating an ellipse about its minor axis. Where this shape is used by a country as the surface for
its mapping system, it is termed the reference ellipsoid. Figure 1.1 illustrates the relationship of the
above surfaces.
The majority of engineering surveys are carried out in areas of limited extent, in which case the
reference surface may be taken as a tangent plane to the geoid and the rules of plane surveying
used. In other words, the curvature of the Earth is ignored and all points on the physical surface are
orthogonally projected onto a flat plane as illustrated in Figure 1.2. For areas less than 10 km
square the assumption of a flat Earth is perfectly acceptable when one considers that in a triangle
of approximately 200 km2
, the difference between the sum of the spherical angles and the plane
angles would be 1 second of arc, or that the difference in length of an arc of approximately 20 km
on the Earth’s surface and its equivalent chord length is a mere 10 mm.
Fig. 1.1
Physical surface
Geoid
EllipsoidA
ξ
Normal (to the
ellipsoid)
Vertical to the geoid
(direction of gravity)

Basic concepts of surveying 3
C
B
A
B′
C′
A′
Fig. 1.2 Projection onto a plain surface
The above assumptions of a flat Earth are, however, not acceptable for elevations as the geoid
would deviate from the tangent plane by about 80 mm at 1 km or 8 m at 10 km. Elevations are
therefore referred to the geoid or MSL as it is more commonly termed. Also, from the engineering
point of view, it is frequently useful in the case of inshore or offshore works to have the elevations
related to the physical component with which the engineer is concerned.
An examination of Figure 1.2 clearly shows the basic surveying measurements needed to locate
points A, B and C and plot them orthogonally as A′, B′ and C′. In the first instance the measured
slant distance AB will fix the position of B relative to A. However, it will then require the vertical
angle to B from A, in order to reduce AB to its equivalent horizontal distance A′B′ for the purposes
of plotting. Whilst similar measurements will fix C relative to A, it requires the horizontal angle
BAC (B′A′C′) to fix C relative to B. The vertical distances defining the relative elevation of the
three points may also be obtained from the slant distance and vertical angle (trigonometrical
levelling) or by direct levelling (Chapter 2) relative to a specific reference datum. The five
measurements mentioned above comprise the basis of plane surveying and are illustrated in Figure
1.3, i.e. AB is the slant distance, AA′ the horizontal distance, A′B the vertical distance, BAA′ the
vertical angle (α) and A′AC the horizontal angle (θ).
It can be seen from the above that the only measurements needed in plane surveying are angle
and distance. Nevertheless, the full impact of modern technology has been brought to bear in the
acquisition and processing of this simple data. Angles are now easily resolved to single-second
accuracy using optical and electronic theodolites; electromagnetic distance measuring (EDM)
A′
B
θ
α
C
A
Fig. 1.3 Basic measurements

equipment can obtain distances of several kilometres to sub-millimetre precision; lasers and north-
seeking gyroscopes are virtually standard equipment for tunnel surveys; orbiting satellites and
inertial survey systems, spin-offs from the space programme, are being used for position fixing off
shore as well as on; continued improvement in aerial and terrestrial photogrammetric equipment
and remote sensors makes photogrammetry an invaluable surveying tool; finally, data loggers and
computers enable the most sophisticated procedures to be adopted in the processing and automatic
plotting of field data.
1.3 CONTROL NETWORKS
The establishment of two- or three-dimensional control networks is the most fundamental operation
in the surveying of an area of large or small extent. The concept can best be illustrated by considering
the survey of a relatively small area of land as shown in Figure 1.4.
The processes involved in carrying out the survey can be itemized as follows:
(1) A careful reconnaissance of the area is first carried out in order to establish the most suitable
positions for the survey stations (or control points) A, B, C, D, E and F. The stations should be
intervisible and so positioned to afford easy and accurate measurement of the distances between
them. They should form ‘well-conditioned’ triangles with all angles greater than 45°, whilst the
sides of the triangles should lie close to the topographic detail to be surveyed. If this procedure
is adopted, the problems of measuring up, over or around obstacles, is eliminated.
The survey stations themselves may be stout wooden pegs driven well down into the ground,
with a fine nail in the top accurately depicting the survey position. Alternatively, for longer life,
D′
DE
F
C
B
A F e n c e
20
40
60
House
F e n c e
F
e
n
c
e
R
O
A
D
H
E
D
G
E
Fig. 1.4 Linear survey

concrete blocks may be set into the ground with some form of fine mark to pinpoint the survey
position.
(2) The distances between the survey stations are now obtained to the required accuracy. Steel
tapes may be laid along the ground to measure the slant lengths, whilst vertical angles may be
measured using hand-held clinometers or Abney levels to reduce the lengths to their horizontal
equivalents. Alternatively, the distances may be measured in horizontal steps as shown in
Figure 1.5. The steps are short enough to prevent sag in the tape and their end positions at 1,
2 and B fixed using a plumb-bob and an additional assistant. The steps are then summed to give
the horizontal distances.
Thus by measuring all the distances, relative positions of the survey stations are located at
the intersections of the straight lines and the network possesses shape and scale. The surveyor
has thus established in the field a two-dimensional horizontal control network whose nodal
points are positioned relative to each other. It must be remembered, however, that all measurements,
no matter how carefully carried out, contain error. Thus, as the three sides of a triangle will
always plot to give a triangle, regardless of the error in the sides, some form of independent
check should be introduced to reveal the presence of error. In this case the horizontal distance
from D to a known position D′ on the line EC is measured. If this distance will not plot
correctly within triangle CDE, then error is present in one or all of the sides. Similar checks
should be introduced throughout the network to prove its reliability.
(3) The proven network can now be used as a reference framework or huge template from which
further measurements can now be taken to the topographic detail. For instance, in the case of
line FA, its position may be physically established in the field by aligning a tape between the
two survey stations. Now, offset measurements taken at right angles to this line at known
distances from F, say 20 m, 40 m and 60 m, will locate the position of the hedge. Similar
measurements from the remaining lines will locate the position of the remaining detail.
The method of booking the data for this form of survey is illustrated in Figure 1.6. The centre
column of the book is regarded as the survey line FA with distances along it and offsets to the
topographic detail drawn in their relative positions as shown in Figure 1.4.
Note the use of oblique offsets to more accurately fix the position of the trees by intersection,
thereby eliminating the error of estimating the right angle in the other offset measurements.
The network is now plotted to the required scale, the offsets plotted from the network and the
relative position of all the topographic detail established to form a plan of the area.
(4) As the aim of this particular survey was the production of a plan, the accuracy of the survey is
governed largely by the scale of the plan. For instance, if the scale was, say, 1 part in 1000, then
a plotting accuracy of 0.1 mm would be equivalent to 100 mm on the ground and it would not
be economical or necessary to take the offset measurements to any greater accuracy than this.
However, as the network forms the reference base from which the measurements are taken, its
position would need to be fixed to a much greater accuracy.
A
1
2
B
Fig. 1.5 Stepped measurement

The above comprises the steps necessary in carrying out this particular form of survey, generally
referred to as a linear survey. It is naturally limited to quite small areas, due to the difficulties of
measuring with tapes and the rapid accumulation of error involved in the process. For this reason
it is not a widely used surveying technique. It does, however, serve to illustrate the basic concepts
of all surveying in a simple, easy to understand manner.
Had the area been much greater in extent, the distances could have been measured by EDM
equipment; such a network is called a trilateration. A further examination of Figure 1.4 shows that
the shape of the network could be established by measuring all the horizontal angles, whilst its
scale or size could be fixed by a measurement of one side. In this case the network would be called
a triangulation. If all the sides and horizontal angles are measured, the network is a triangulateration.
Finally, if the survey stations are located by measuring the adjacent angles and lengths shown in
Figure 1.7, thereby constituting a polygon A, B, C, D, E, F, the network is a traverse. These then
constitute all the basic methods of establishing a horizontal control network, and are dealt with in
more detail in Chapter 6.
1.4 LOCATING POSITION
The method of locating the position of topographic detail by right-angled offsets from the sides of
the control network has been mentioned above. However, this method would have errors in establishing
Fence Fence Page 4
B
C
E
E
F
Wood
constr.
HEDGE
1.90
5.20
2.85
60.00
52.30
43.60
40.00
31.00
20.00
12.50
A
84.50
9.25
9.10
6.8
7.1
10.306.30
6.30 10.30
Fence
Fence
Page 3
BARN
6.54
Fig. 1.6 Field book

the line FA, in setting out the right angle (usually by eye) and in measuring the offset. It would
therefore be more accurate to locate position directly from the survey stations. The most popular
method of doing this is by polar coordinates as shown in Figure 1.8. A and B are survey stations of
known position in a control network, from which the measured horizontal angle BAP and the
horizontal distance AP will fix the position of point P. There is no doubt that this is the most popular
method of fixing position, particularly since the advent of EDM equipment. Indeed, the method of
traversing is a repeated application of this process.
An alternative method is by intersection where P is fixed by measuring the horizontal angles BAP
and ABP as shown in Figure 1.9. This method forms the basis of triangulation. Similarly, P may be
fixed by the measurement of horizontal distances AP and BP and forms the basis of the method of
LBC
A
c
d
ba
LAB
LCD
LDE
LEF
LFA
E
F
D
C
B
Fig. 1.7 Traverse
C
A
Building (plan view)
P1
D1
P2
D3
P3
Control network
B
To D
α1
D2
Fig. 1.8 Polar coordinates

BA
P
Fig. 1.9 Intersection
trilateration. In both these instances there is no independent check as a position for P (not necessarily
the correct one) will always be obtained. Thus at least one additional measurement is required
either by combining the angles and distances (triangulateration) by measuring the angle at P as a
check on the angular intersection, or by producing a trisection from an extra control station.
The final method of position fixing is by resection (Figure 1.10). This is done by observing the
horizontal angles at P to at least three control stations of known position. The position of P may be
obtained by a mathematical solution as illustrated in Chapter 6.
Once again, it can be seen that all the above procedures simply involve the measurement of angle
and distance.
1.5 LOCATING TOPOGRAPHIC DETAIL
Topographic surveying of detail is, in the first instance, based on the established control network.
The accurate relative positioning of the control points would generally be by the method of traversing
or a combination of triangulation and trilateration (Chapter 6). The mean measured angles and
distances would be processed, to provide the plane rectangular coordinates of each control point.
Each point would then be carefully plotted on a precisely constructed rectangular grid. The grid
would be drawn with the aid of a metal template (Figure 1.11), containing fine drill holes in an
exact grid arrangement. The position of the holes is then pricked through onto the drawing material
using the precisely fitting punch shown. Alternatively, the grid would be drawn using a computer-
driven coordinatorgraph on a flat-bed or drum plotter. The topographic detail is then drawn in from
the plotted control points which were utilized in the field.
Fig. 1.10 Resection
B
CA
P

1.5.1 Field survey
In the previous section, the method of locating detail by offsets was illustrated. In engineering
surveys the more likely method is by polar coordinates, i.e. direction relative to a pair of selected
control points, plus the horizontal distance from one of the known points, as shown in Figure 1.8.
The directions would be measured by theodolite and the distance by EDM, to a detail pole held
vertically on the detail (Figure 1.12); hence the ideal instrument would be the electronic tacheometer
or total station.
The accuracy required in the location of detail is a function of the scale of the plan. For instance,
if the proposed scale is 1 in 1000, then 1mm on the plan would represent 1000 mm on the ground.
If the plotting accuracy was, say, 0.2 mm, then the equivalent field accuracy would be 200 mm and
distance need be measured to no greater accuracy than this. The equivalent angular accuracy for a
length of sight at 200 m would be about 3′ 20′′. From this it can be seen that the accuracy required
to fix the position of detail is much less than that required to establish the position of control points.
It may be, depending on the scale of the plan and the type of detail to be located, that stadia
tacheometry could be used for the process, in the event of there being no other alternative.
The accuracy of distance measurement in stadia tacheometer (D = 100 × S cos2
θ), as shown in
Chapter 2, is in the region of 1 in 300, equivalent to 300 mm in an observation distance of 100 m.
Thus before this method can be considered, the scale of the plan must be analysed as above, the
average observation distance should be considered and the type of detail, hard or soft, reconnoitred.
Even if all these considerations are met, it must be remembered that the method is cumbersome and
uneconomical unless a direct reading tacheometer is available.
1.5.2 Plotting the detail
The purpose of the plan usually defines the scale to which it is plotted. The most common scale for
construction plans is 1 in 500, with variations above or below that, from 1 in 2500 to 1 in 250.
The most common material used is plastic film with such trade names as ‘Permatrace’. This is an
Fig. 1.11 Metal template and punch

Fig. 1.12 ‘Detail pole’ locating topographic detail
extremely durable material, virtually indestructible with excellent dimensional stability. When the
plot is complete, paper prints are easily obtained.
Although the topographic detail could be plotted using a protractor for the direction and a scale
for the distances, in a manner analogous to the field process, it is a trivial matter to produce ‘in-
house’ software to carry out this task. Using the arrangement shown in Figure 1.13, the directions
and distances are input to the computer, changed to two-dimensional coordinates and plotted direct.
A simple question asks the operator if he wishes the plotted point to be joined to the previous one
and in this way the plot is rapidly progressed. This elementary ‘in-house’ software simply plots
points and lines and the reduced level of the points, where the vertical angle is included. However,
there is now an abundance of computer plotting software available that will not only produce a
contoured plot, but also supply three-dimensional views, digital ground models, earthwork volumes,
road design, drainage design, digital mapping, etc.
1.5.3 Computer systems
To be economically viable, practically all major engineering/surveying organizations use an automated
plotting system. Very often the total station and data logger are purchased along with the computer
hardware and software, as a total operating system. In this way interface and adaptation problems
are precluded. Figure 1.14 shows such an arrangement including a ‘mouse’for use on the digitizing
tablet. An AO flat-bed plotter is networked to the system and located separately.
The essential characteristics of such a system are:
(1) Capability to accept, store, transfer, process and manage field data that is input manually or
directly from an interfaced data logger (Figure 1.15).
(2) Software and hardware to be in modular form for easy accessing.
(3) Software to use all modern facilities, such as ‘windows’, different colour and interactive screen
graphics, to make the process user friendly.
(4) Continuous data flow from field data to finished plan.

Fig. 1.13 Computer driven plotter
Fig. 1.14 Computer system with digitizing tablet
(5) Appropriate data-base facility, for the storage and management of coordinate and cartographic
data necessary for the production of digital ground models and land/geographic information
systems.
(6) Extensive computer storage facility.
(7) High-speed precision flat-bed or drum plotter.

To be truly economical, the field data, including appropriate coding of the various types of detail,
should be captured and stored by single-key operation, on a data logger interfaced to a total station.
The computer system should then permit automatic transfer of this data by direct interface between
the logger and the system. The modular software should then: store and administer the data; carry
out the mathematical processing, such as network adjustment, production of coordinates and elevations;
generate data storage banks; and finally plot the data on completion of the data verification process.
Prior to plotting, the data can be viewed on the screen for editing purposes. This can be done from
the keyboard or by light pen on the screen using interactive graphics routines. The plotted detail can
be examined, moved, erased or changed, as desired. When the examination is complete, the command
to plot may then be activated. Figure 1.16 shows an example of a computer plot.
1.5.4 Digital ground model (DGM)
A DGM is a three-dimensional, mathematical representation of the landform and all its features,
stored in a computer data base. Such a model is extremely useful in the design and construction
process, as it permits quick and accurate determination of the coordinates and elevation of any
point.
The DGM is formed by sampling points over the land surface and using appropriate algorithms
to process these points to represent the surface being modelled. The methods in common use are
modelling by ‘strings’, ‘regular grids’ or ‘triangular facets’. Regardless of the methods used, they
will all reflect the quality of the field data.
A ‘string’ comprises a series of points along a feature and so such a system stores the position of
features surveyed. It is widely used for mapping purposes due to its flexibility, its accuracy along
the string and its ability to process large amounts of data very quickly. However, as it does not store
the relationship between strings, a searching process is essential when the levels of points not
Fig. 1.15 Data logger

Fig. 1.16 Computer plot
included in a string are required. Thus its weakness lies in the generation of accurate contours and
volumes.
The ‘regular grid’ method uses appropriate algorithms to convert the sampled data to a regular
grid of levels. If the field data permit, the smaller the grid interval, the more representative of
landform it becomes. Although a simple technique, it only provides a very general shape of the
landform, due to its tendency to ignore vertical breaks of slope. Volumes generated also tend to be
rather inaccurate.
In the ‘triangular grid’ method, ‘best fit’ triangles are formed between the points surveyed. The
ground surface therefore comprises a network of triangular planes at various angles (Figure 1.17(a)).
Computer shading of the model (Figure 1.17(b)) provides an excellent indication of the landform.
In this method vertical breaks are forced to form the sides of triangles, thereby maintaining correct
ground shape. Contours, sections and levels may be obtained by linear interpolation through the
triangles. It is thus ideal for contour generation (Figure 1.18) and highly accurate volumes. The
volumes are obtained by treating each triangle as a prism to the depth required; hence the smaller
the triangle, the more accurate the final result.
1.5.5 Computer-aided design (CAD)
In addition to the production of DGMs and contoured plans, the modern computer surveying
system permits the easy application of the designed structure to the finished plan. The three-

dimensional information held in the data base supplies all the ground data necessary to facilitate the
finished design. Figure 1.19 illustrates its use in road design.
The environmental impact of the design can now be more readily assessed by producing perspective
views as shown in Figures 1.20(a) and (b). The new environmental impact laws make this latter
tool extremely valuable.
1.5.6 Land/geographic information systems (LIS/GIS)
Prior to the advent of computers, land-related information was illustrated by means of overlay
tracings on the basic topographic map or plan. For instance, consider a plan of an urban area on
which it is also required to show the public utilities, i.e. the gas mains, electrical cables, substations,
drainage system, manholes, etc. As adding all this information to the base plan would render it
completely unreadable, each system was drawn on separate sheets of tracing paper. Each tracing
could then be overlain over the base plan, as and when required (Figure 1.21). In addition, a large
ledger was kept, as part of the arrangement, itemizing the dimensions of the pipes, the material
used, the ownership, the condition, the ownership of the land under which it passed, etc. All this
information, used with the base plan and overlays, comprised a cumbersome land information system.
(a) (b)
Fig. 1.17 (a) Triangular grid model, and (b) Triangular grid model with computer shading
Fig. 1.18 Computer generated contour model

All this information and more can now be stored in a computer to form the basis of the modern-
day L/GIS. Thus a L/GIS is a land-related data base held in a highly structured form within the
computer, in order to make it easier to manage, update, access, interrogate and retrieve. Although
many sophisticated commercial packages are available, the process is still in a state of evolution.
The ultimate GIS is one which could supply all the information relating to land from, say, 10 km
above its surface to 100 km below; the amount of information to be stored is almost incomprehensible.
It may be necessary to consider land boundaries, areas of land, type of soil, erosion characteristics,
type of property, ownership, street names, rateable values, landslip data, past and future land use,
agricultural areas, flood protection, mineral resources, public utilities; the list is inexhaustible. In
addition, all this information must be related to good-quality large-scale maps or plans. Further to
this, there is the problem of different individuals wishing to access the system for their own
Fig. 1.19 Computer-aided road design
(a) (b)
Fig. 1.20 Perspectives with computer shading

requirements. There is the private landowner wishing to know about future land use, the planners,
the local authority administrators, the civil engineer, the mineral operator, the lawyer, all requiring
rapid and easy access to the information specific to their needs. The system would thereby improve
the administration of all legal matters appertaining to land, furnish data for the better administration
of the land, facilitate resource management and environmental planning, etc.
The problems of producing an efficient L/GIS are complex and numerous. The information must
be efficiently filed, uniquely coded, conveniently stored, easily accessed, interrogated and retrieved,
and highly flexible in its applications.
The first problem is the availability of good-quality large-scale plans on an approved coordinate
system. This can be achieved by surveying the areas concerned or, where acceptable plans are
available, digitizing them. A system of quality control is necessary to ensure a common standard
from all the sources.
A system of identifying and indexing the various land parcels is then necessary, based in the first
instance on the coordinate system used.
When the topographic structure is in place for on-screen analysis and hard copy availability, the
massive problem of finding, checking, proving and storing the large volume of land-related data
follows. It may be necessary to layer this information in files within the data base and combine this
with powerful data-base management software to ensure its efficient manipulation. The coding
process is far more complex than the surveyor is normally used to. In surveying an area, for
instance, the surveyor is concerned essentially with the shape, size and position of a feature.
Therefore, if surveying a number of buildings, a simple code of B1, B2, etc. may be used, i.e. B for
Building, the number denoting the number of buildings. In a L/GIS system, not only is the above
information required, but it is necessary to know the type of building (office, residential, industrial,
etc.), the mode of construction (brick or concrete), the number of storeys, the ownership, the
present occupancy, the specific use, the rateable value, etc. Thus it can be seen that the coding is
an extremely complex issue. The situation may be further complicated by the problem of confidentiality,
for whilst the system should be user friendly, it should not be possible to access confidential data.
Integration of all sources of data may be rendered extremely difficult, if not impossible, by the
attitudes of the various institutions holding the information.
It can be seen that the problems of producing a multi-purpose land information system are
complex. In the case of a geographic information system these problems are magnified. The GIS
is similarly concerned with the storage, management and analysis of spatially related data, but on
a much greater scale. The ultimate GIS would be a global information system. The geographic
information could be necessary for such processes as weather forecasting, flood forecasting from
rainfall records, stream and river location, drainage patterns and systems, position and size of dams
Overlays
E
D
C
B
A – Base plan
F
Fig. 1. 21 The concept of a L/GIS: B, gas pipes; C, electric cables; D, drainage system, etc.

and reservoirs, land use and transportation patterns over very wide areas; once again the list is
inexhaustible.
Thus, although the formation of a L/GIS is a formidable problem, the necessity for an efficient
and accurate source of land-related data makes it mandatory as a powerful land management tool.
As good-quality plans form the basis of such a system, it is feasible that surveyors, who are the
experts in measurement and position, should play a prominent part in the design and management
of such systems.
1.5.6.1 GIS data
From the broad introduction given to GIS it can be seen that a GIS is a computer-based system for
handling not only physical location but also the attributes associated with that location. It thus
possesses a graphical display in two or three dimensions of the spatial data, combined with a
database for the non-spatial data, i.e. the attribute information. The prime aspect in the construction
of a GIS is the acquisition, from many different sources, conversion and entry of the data.
The spatial data may be acquired from a variety of sources: from digitized maps and plans, from
aerial photographs, from satellite imagery, or directly from GPS surveys. However, in order to
represent this complex, three-dimensional reality in a spatial database, it is modelled using points,
lines, areas, surfaces and networks. For instance, if we consider an underground drainage system,
the pipes would be represented by lines; the manhole positions by points; the parcels of land
ownership forming closed boundaries whose polygon shape is defined by coordinates would be
represented by areas; whilst the three-dimensional land surface through which the pipes pass would
be represented by a surface. Such a GIS would probably incorporate a network, which represents
the whole branching system of pipes (line segments), and is used to simulate flow through the pipes
or indicate the buildings affected by a break in the pipe network at a specific point. The attributes
attached to this network, such as type and size of pipe, depth below ground, rate of flow, gradients,
etc., would be stored in the associated database. The linking of the spatially referenced data with
their attributes is the basis of GIS.
The above features can be represented within a GIS in either vector or raster format; their
relative spatial relationships are given by their topology (Figure 1.22).
Vector data uses dots and lines, similar to the plotting of x, y coordinates on a plan, and the
joining up of those coordinated points with lines and curves to give shape and position. The vector
format provides an accurate representation of the spatially referenced data incorporating the topology
and other spatial relationships between the individual entries.
Scrub
Marsh
Wood land
(c)(a) (b)
Marsh
Scrub
Wood land
Fig. 1.22 (a) Shows standard topographic plan. (b) shows vector representation of (a). (c) Shows raster
representation of (a)
W W W W W W W W W W W W W W W W
W W W W W W W W W W W W W S S S
W W W W W W W W W W W W S S S S
W W W W W W W W W W W S S S S S
W W W W W W W W W W S S S S S S
W W W W W W W W M M M S S S S S
M M M M M M M M M M M M S S S S
M M M M M M M M M M M M S S S S
M M M M M M M M M M M M M S S S
M M M M M M M M M M M M M S S S
M M M M M M M M M M M M M M S S
M M M M M M M M M M M M M M S S

The GIS vector model differs from that of CAD or simple drawing packages as each dot (called
vertices in GIS), line segment, area or polygon is uniquely identified and their relationships stored
in the database. Computer data storage is very economical, but certain analytical processes have
high computational requirements resulting in slow operations or the use of high specification
hardware. The vector data model is ideally suited to the representation of linear networks such as
roads, railways and pipelines. It also provides accurate measurement of areas and lengths. It is the
obvious format for inputting digital data obtained by conventional survey procedures or by digitizing
existing plans or maps.
The raster format uses pixels (derived from ‘picture elements’) or grid cells. It is not as accurate
or flexible as vector format as each coordinate may be represented by a cell and each line by an
array of cells. Thus data can be positioned only to the nearest grid cell. Examples of data in the
raster format are aerial photographs, satellite imagery and scanned maps or plans. An example from
reality would be moorland comprising areas of marsh and scrub, etc., where the vague boundaries
would not be unduly affected by the inaccuracy of the format presentation. In addition to producing
a coarse resolution of the data, each cell contains a single value representing the attribute contained
within the area of the cell. The resolution of the data may be improved using a smaller cell size, but
this would increase the computer storage, which tends, in any case, to be uneconomical using the
raster format. The computer finds it easier to collect, store and manage raster data using such
techniques as overlay, buffering and network analysis.
The above are the two main data models, but a third object-based model is available which
represents the data as it appears in the real world, thereby making it easier to understand (Figure 1.20).
It does, however, result in very high processing requirements.
Initially, GIS systems used one format or the other. However, modern GIS software permits
conversion between the two and can display vector data over the top of raster data.
In all GIS systems, the data is layered. For instance, one layer would contain all the houses in the
area, another layer all the water pipes, and so on, as shown in Figure 1.21. This allows data to be
shown separately but still retains cross-referencing between the layers for analyses or interrogation.
All the layers are interrelated and to a common scale so that they can be accurately overlain.
1.5.6.2 Topology
Topology is a branch of mathematics dealing with the relative relationships between individual
entities. It is a method of informing the computer how to arrange the data input into its correct
relative position. Important topological concepts are:
• Adjacency: consider a line defining the edge of a road: on which side of that line does the road
lie?
• Connectivity: which points must be connected to show each side of the road?
• Orientation: defines the starting point and ending point in a chain of points describing the road?
• Nestedness: what spatial objects, such houses, lie within a given polygon, such as its property
boundaries?
Once these concepts are placed in the computer data files the relative relationships, or topology, of
the spatial data can be realized.
1.5.6.3 Functionality
The GIS is not just a simple graphic display of spatial data or of attribute data, but a system
combining both to provide sophisticated functions that assist management and decision making.
The first and most important step is the acquisition and input of data. It is important because the

GIS is only as good as the data provided. The data may be obtained from many sources already
mentioned, such as the digitization of existing graphic material; the scanning of topographic maps/
plans; aerial photographs (or the photographs of satellite imagery); keyboard entry of survey data,
attribute data or direct interface of GPS data; all of which must be transformed, where necessary,
into digital form. In addition, it may be possible to use existing digital data sets.
The data is not only sorted within the computer, but is indexed and managed to ensure controlled
and co-ordinated access. The data must be structured in such a way as to ensure the reliability,
security and integrity of the data.
The GIS provides links between spatial and non-spatial data, allowing sophisticated analysis of
the total data set. Interrogation may be graphics-driven or data-driven and require the selective
display of spatial and non-spatial data. Examples of the more common spatial analysis and
computational functions are illustrated below.
• Buffering involves the creation of new polygons or buffer zones around existing nodes or points
at set intervals. An example may be a break in a water pipe: a buffer zone may be created around
that point showing the area which may be flooded. Similarly, the creation of buffers around a
source of contamination, indicating the various areas of intensity of contamination.
• Overlay is the process of overlaying spatial data of one type onto another type. For instance, the
overlaying of soil type data on drainage patterns may indicate the best positions to site land
drains.
• Network analysis may be used to simulate traffic flows through a network of streets in a busy
urban complex in order to optimize and improve traffic conditions.
• Terrain analysis could involve the creation of a three-dimensional ground model in order to
investigate the environmental impact of a proposed construction, for instance.
• Contouring is the connection of points of equal value to form lines. These could be points of
elevation to give ground contours, or points of a particular attribute to, perhaps, give population
density contour lines.
• Area and length calculations is largely self-explanatory and could involve the area of derelict
land for future housing development, or lengths of highway to be widened.
All these functions can be viewed on the screen, or output in the form of plans, graphs, tables or
reports.
The use of GIS, therefore, removes the need for paper plans and associated documents and
greatly speeds up operations as the data, both spatial and non-spatial, can be rapidly updated, edited
and transferred to other computers networked to the central GIS. It thus has the advantages of
transferring data between multiple users, thereby minimizing duplication and increasing security
and reliability of the data. Specific scenarios can be modelled to test possible outcomes and create
better-informed decision making. For instance, using various layers of data such as drainage patterns,
surface and sub-soil data, ground slopes, and rainfall values, areas of potential erosion or landslip
can be identified. Thus the GIS not only provides effective data management and analysis, but also
allows spatial features and their relationships to be visualized. In this way planning and investment
decisions can be made with confidence.
1.5.6.4 Applications of GIS
GIS can be applied in any situation where spatially referenced data requires modelling, analysis and
management. Some examples are:
Facilities management Organizations such as those dealing with gas, water, electricity or sewerage
are responsible for vast amounts of pipelines, cables, tunnels, buildings and land, all of which

require monitoring, maintenance and management in order to give an efficient and effective service
to customers.
Highways maintenance This situation is very similar to the above but deals with roads, motorways,
bridges, road furniture, etc., all of which is spatially referenced and requires maintenance and
management. Three-dimensional ground models can be used for design and environmental impact
studies.
Housing associations These organizations are responsible for the building, maintenance, leasing,
renting or sale of houses on a massive scale. Not only is the geographic distribution of the properties
required, but full details of the properties are also vital. To assist in operational management and
strategic planning such information as rent arrears and the geographic clustering; housing types;
properties sold, leased or rented; conditions/repairs; population trends; development sites; bad debt
hotspots – the list is endless. Thus paper-based land terriers are replaced, there is high-quality
visual representation of spatial data, improved productivity and more efficient management tools.
The above examples clearly illustrate the importance of GIS and the manner of its application.
Other areas which would benefit from its use are environmental management; transportation;
market analysis using, say, socio-economic population distribution patterns; and land use patterns.
Indeed, wherever the relationship and interaction of various spatially referenced data is required,
GIS provides a powerful analytical tool.
1.5.7 Laser scanner
Laser scanning, in a terrestrial or airborne form, is a relatively new and powerful surveying technique.
The system provides 3-D location of features and surfaces quickly and accurately, in real time if
necessary.
The system is a combined hardware and software package. The hardware consists of a tripod-
mounted pulsed laser range finder and a mechanical scanner. The time taken by the laser pulse to
hit the target and return is measured by the picosecond timing circuitry of the unit’s signal detector,
and the range calculated. The amount of energy reflected by the target surface is a function of the
target’s characteristics, such as roughness, colour, etc. The amplitude of the returned pulse gives an
intensity or brightness value. A Class 1, eye safe laser, operating in the near-infrared region at
0.9 µm is used, with an operating range of 0.1–350m and a beam width of about 300mm at 100m
distance. The scanning density can be altered and set in increments of 0.25°, 0.5° and 1°. A rotating
polygonal mirror directs the laser beam in the horizontal and vertical directions. Angle encoders
record the orientation of the mirror. Thus, each point within the raster image of range and intensity
is accurately positioned in 3-D and illustrated via the controlling laptop PC. Data can be acquired
at rates as high as 6000 measurements per second using a laser pulsing at 20 kHz, with accuracies
of ±5 mm. In some systems, using special targets other than the actual ground or structure surfaces,
accuracies of ±2 mm are achievable. If the tripod is set over a point of known coordinates and
orientated into the coordinate system in use, then the spatial position of the points scanned can be
defined in that system. At the present time the laser scanning device can vary in weight from
13.5 kg to 30 kg, depending on the make of the unit. One particular unit incorporates a colour CCD
camera to capture scenes for later analysis. This latter point indicates the many and varied ways in
which modern technology is being utilized in spatial data capture.
The laser device is controlled and the data processed by means of a PC connected to it through
serial and parallel cables. The scanner parameters are set by the operator and the data downloaded
in real time for 3-D screen viewing. The raster style 3-D picture can be rotated in space for viewing
from any angle as scanning takes place. The range to points can be queried and inter-distances
between points measured. The screen image enables the operator to evaluate the quality of the data
and, if necessary, change the parameter settings or move the scanner to a better site position. If the

survey area is extensive, reflectors may be used in the scanned portions to allow the co-ordination
and merging of various scans. The intensities of the laser signals, which in effect describe the
characteristics of the points in question, may be illustrated on the screen using different colours,
thereby highlighting variations in the data. The data files are naturally quite large, and a figure
quoted for the survey of a room area of 30 m2
with pillars and windows, was 2 Mb. For best results
the field data can be transferred to a more powerful graphics workstation for further processing,
editing and analysis. Precise 2-D drawings with elevations, or 3-D models can be generated.
Applications of this revolutionary system occur in all aspects of surveying, mining and civil
engineering. It is particularly useful in inaccessible locations such as building facades, mine and
quarry faces, and areas which are unsafe such as cliff faces, airport runways, busy highways and
hazardous areas in chemical and nuclear installations. The applications mentioned are those that are
particularly difficult for conventional surveying procedures. However, this does not preclude its
use in all those areas of conventional survey, including tunnelling.
The principles outlined above can also be used in airborne situations where the aircraft equipped
with GPS is positioned in space by a single ground-based GPS station and an inertial navigation
unit is used for the determination of roll, pitch and yaw. In this way the position and attitude of the
scanner is fixed in the GPS coordinate system (WGS84), and so also are the terrain positions.
Transformation to a local reference system will also require a geoid model.
The flying height varies from 300–1000 m, with the laser beam scanning at a rate as high as
25000 pulses per second across a swath beneath the aircraft.
At the present time, ground-based systems are large, heavy and expensive, but there is no doubt
that, within a very short period by time, they will become smaller, more sophisticated, and a major
method of 3-D detailing.
1.6 SUMMARY
In the preceding sections an attempt has been made to outline the basic concepts of surveying.
Because of their importance they will now be summarized as follows:
(1) Reconnaissance is the first and most important step in the surveying process. Only after a
careful and detailed reconnaissance of the area can the surveyor decide upon the techniques
and instrumentation required to economically complete the work and meet the accuracy
specifications.
(2) Control networks not only form a reference framework for locating the position of topographic
detail and setting out constructions, but may also be used as a base for minor control networks
containing a greater number of control stations at shorter distances apart and to a lower order
of accuracy, i.e. a, b, c, d in Figure 1.7. These minor control stations may be better placed for
the purpose of locating the topographic detail.
This process of establishing the major control first to the highest order of accuracy, as a
framework on which to connect the minor control, which is in turn used as a reference framework
for detailing, is known as working from the whole to the part and forms the basis of all good
surveying procedure.
(3) Errors are contained in all measurement procedures and a constant battle must be waged by the
surveyor to minimize their effect.
It follows from this that the greater the accuracy specifications the greater the cost of the
survey for it results in more observations, taken with greater care, over a longer period of time,
using more precise (and therefore more expensive) equipment. It is for this reason that major

control networks contain the minimum number of stations necessary and surveyors adhere to
the economic principle of working to an accuracy neither greater than nor less than that required.
(4) Independent checks should be introduced not only into the field work, but also into the subsequent
computation and reduction of field data. In this way, errors can be quickly recognized and dealt
with.
Data should always be measured more than once. Examination of several measurements will
generally indicate the presence of blunders in the measuring process. Alternatively, close
agreement of the measurements is indicative of high precision and generally acceptable field
data, although, as shown later, high precision does not necessarily mean high accuracy, and
further data processing may be necessary to remove any systematic error that may be present.
(5) Commensurate accuracy is advised in the measuring process, i.e. the angles should be measured
to the same degree of accuracy as the distances and vice versa. The following rule is advocated
by most authorities for guidance: 1′′ of arc subtends 1 mm at 200 m. This means that if distance
is measured to, say, 1 in 200 000, the angles should be measured to 1′′ of arc, and so on.
(6) The model used to illustrate the concepts of surveying is limited in its application and for most
engineering surveys may be considered obsolete. Nevertheless it does serve to illustrate those
basic concepts in simple, easily understood terms, to which the beginner can more easily relate.
In the majority of engineering projects, sophisticated instrumentation such as ‘total stations’interfaced
with electronic data loggers is the norm. In some cases the data loggers can directly drive plotters,
thereby producing plots in real time.
Further developments are in the use of satellites to fix three-dimensional position. Such is the
accuracy and speed of positioning using the latest GPS satellites that they may be used to establish
control points, fix topographic detail, set out position on site and carry out continuous deformation
monitoring. Indeed, in the very near future, the use of networks may be of purely historical interest.
Also, inertial positioning systems (IPS) provide a continuous output of position from a known
starting point, independent of any external agency, environmental conditions or location. Integration
of GPS and IPS may provide a formidable positioning process in the future.
However, regardless of the technological advances in surveying, attention must always be given
to instrument calibration, carefully designed projects and meticulous observation. As surveying is
essentially the science of measurement, it is necessary to examine the measured data in more detail,
as follows.
1.7 UNITS OF MEASUREMENT
The system most commonly used in the measurement of distance and angle is the ‘Systeme
Internationale’, abbreviated to SI. The basic units of prime interest are:
Length in metres (m)
from which we have:
1 m = 103
millimetres (mm)
1 m = 10–3
kilometres (km)
Thus a distance measured to the nearest millimetre would be written as, say, 142.356 m.
Similarly for areas we have:
1 m2
= 106
mm2

104
m2
= 1 hectare (ha)
106
m2
= 1 square kilometre (km2
)
and for volumes, m3
and mm3
.
There are three systems used for plane angles, namely the sexagesimal, the centesimal and
radiants (arc units).
The sexagesimal units are used in many parts of the world, including the UK, and measure angles
in degrees (°), minutes (′) and seconds (′′) of arc, i.e.
1° = 60′
1′ = 60′′
and an angle is written as, say, 125° 46′ 35′′.
The centesimal system is quite common in Europe and measures angles in gons (g), i.e.
1 gon = 100 cgon (centigon)
1 cgon = 10 mgon (milligon)
A radian is that angle subtended at the centre of a circle by an arc on the circumference equal in
length to the radius of the circle, i.e.
2π rad = 360° = 400 gon
Thus to transform degrees to radians, multiply by π /180°, and to transform radians to degrees,
multiply by 180°/π. It can be seen that:
1 rad = 57.2957795° = 63.6619972 gon
A factor commonly used in surveying to change angles from seconds of arc to radians is:
α rad = α ′′/206265
where 206265 is the number of seconds in a radian.
Other units of interest will be dealt with where they occur in the text.
1.8 SIGNIFICANT FIGURES
Engineers and surveyors communicate a great deal of their professional information using numbers.
It is important, therefore, that the number of digits used, correctly indicates the accuracy with
which the field data were measured. This is particularly important since the advent of pocket
calculators, which tend to present numbers to as many as eight places of decimals, calculated from
data containing, at the most, only three places of decimals, whilst some eliminate all trailing zeros.
This latter point is important, as 2.00 m is an entirely different value to 2.000 m. The latter number
implies estimation to the nearest millimetre as opposed to the nearest 10 mm implied by the former.
Thus in the capture of field data, the correct number of significant figures should be used.
By definition, the number of significant figures in a value is the number of digits one is certain
of plus one, usually the last, which is estimated. The number of significant figures should not be
confused with the number of decimal places. A further rule in significant figures is that in all
numbers less than unity, the number of zeros directly after the decimal point and up to the first non-
zero digit are not counted. For example:

Two significant figures: 40, 42, 4.2, 0.43, 0.0042, 0.040
Three significant figures: 836, 83.6, 80.6, 0.806, 0.0806, 0.00800
Difficulties can occur with zeros at the end of a number such as 83600, which may have three, four
or five significant figures. This problem is overcome by expressing the value in powers of ten, i.e.
8.36 × 104
implies three significant figures, 8.360 × 104
implies four significant figures and
8.3600 × 104
implies five significant figures.
It is important to remember that the accuracy of field data cannot and should not be improved in
the computational processes to which it is subjected.
Consider the addition of the following numbers:
155.486
7.08
2183.0
42.0058
If added on a pocket calculator the answer is 2387.5718; however, the correct answer with due
regard to significant figures is 2387.6. It is rounded off to the most extreme right-hand column
containing all the significant figures, which in the example is the column immediately after the
decimal point. In the case of 155.486 + 7.08 + 2183 + 42.0058 the answer is 2388. This rule also
applies to subtraction.
In multiplication and division, the answer should be rounded off to the number of significant
figures contained in that number having the least number of significant figures in the computational
process. For instance, 214.8432 × 3.05 = 655.27176, when computed on a pocket calculator;
however, as 3.05 contains only three significant figures, the correct answer is 655. Consider
428.4 × 621.8 = 266379.12, which should now be rounded to 266400 = 2.664 ×105
, which has four
significant figures. Similarly, 41.8 ÷ 2.1316 = 19.609682 on a pocket calculator and should be
rounded to 19.6.
When dealing with the powers of numbers the following rule is useful. If x is the value of the first
significant figure in a number having n significant figures, its pth power is rounded to:
n – 1 significant figures if p ≤ x
n – 2 significant figures if p ≤ 10x
For example, 1.58314
= 8.97679 when computed on a pocket calculator. In this case x = 1, p = 4 and
p ≤ 10x; therefore, the answer should be quoted to n – 2 = 3 significant figures = 8.98.
Similarly, with roots of numbers, let x equal the first significant figure and r the root; the answer
should be rounded to:
n significant figures when rx ≥ 10
n – 1 significant figures when rx < 10
For example:
36
1
2 = 6, because r = 2, x = 3, n = 2, thus rx < 10, and answer is to n – 1 = 1 significant figure.
415.36
1
4 = 4.5144637 on a pocket calculator; however, r = 4, x = 4, n = 5, and as rx > 10, the
answer is rounded to n = 5 significant figures, giving 4.5145.
As a general rule, when field data are undergoing computational processing which involves several
intermediate stages, one extra digit may be carried throughout the process, provided the final
answer is rounded to the correct number of significant figures.

1.9 ROUNDING OFF NUMBERS
It is well understood that in rounding off numbers, 54.334 would be rounded to 54.33, whilst
54.336 would become 54.34. However, with 54.335, some individuals always round up, giving
54.34, whilst others always round down to 54.33. This process creats a systematic bias and should
be avoided. The process which creates a more random bias, thereby producing a more representative
mean value from a set of data, is to round up when the preceding digit is odd but not when it is even.
Using this approach, 54.335 becomes 54.34, whilst 54.345 is 54.34 also.
1.10 ERRORS IN MEASUREMENT
It should now be apparent that position fixing simply involves the measurement of angles and
distance. However, all measurements, no matter how carefully executed, will contain error, and so
the true value of a measurement is never known. It follows from this that if the true value is never
known, the true error can never be known and the position of a point known only within certain
error bounds.
The sources of error fall into three broad categories, namely:
(1) Natural errors caused by variation in or adverse weather conditions, refraction, gravity effects,
etc.
(2) Instrumental errors caused by imperfect construction and adjustment of the surveying instruments
used.
(3) Personal errors caused by the inability of the individual to make exact observations due to the
limitations of human sight, touch and hearing.
1.10.1 Classification of errors
(1) Mistakes are sometimes called gross errors, but should not be classified as errors at all. They
are blunders, often resulting from fatigue or the inexperience of the surveyor. Typical examples
are omitting a whole tape length when measuring distance, sighting the wrong target in a round
of angles, reading ‘6’ on a levelling staff as ‘9’ and vice versa. Mistakes are the largest of the
errors likely to arise, and therefore great care must be taken to obviate them.
(2) Systematic errors can be constant or variable throughout an operation and are generally attributable
to known circumstances. The value of these errors can be calculated and applied as a correction
to the measured quantity. They can be the result of natural conditions, examples of which are:
refraction of light rays, variation in the speed of electromagnetic waves through the atmosphere,
expansion or contraction of steel tapes due to temperature variations. In all these cases, corrections
can be applied to reduce their effect. Such errors may also be produced by instruments, e.g.
maladjustment of the theodolite or level, index error in spring balances, ageing of the crystals
in EDM equipment.
There is the personal error of the observer who may have a bias against setting a micrometer
or in bisecting a target, etc. Such errors can frequently be self-compensating; for instance, a
person setting a micrometer too low when obtaining a direction will most likely set it too low
when obtaining the second direction, and the resulting angle will be correct.
Systematic errors, in the main, conform to mathematical and physical laws; thus it is argued
that appropriate corrections can be computed and applied to reduce their effect. It is doubtful,

however, whether the effect of systematic errors is ever entirely eliminated, largely due to the
inability to obtain an exact measurement of the quantities involved. Typical examples are: the
difficulty of obtaining group refractive index throughout the measuring path of EDM distances;
and the difficulty of obtaining the temperature of the steel tape, based on air temperature
measurements with thermometers. Thus, systematic errors are the most difficult to deal with
and therefore they require very careful consideration prior to, during, and after the survey.
Careful calibration of all equipment is an essential part of controlling systematic error.
(3) Random errors are those variates which remain after all other errors have been removed. They
are beyond the control of the observer and result from the human inability of the observer to
make exact measurements, for reasons already indicated above.
Random variates are assumed to have a continuous frequency distribution called normal
distribution and obey the law of probability. A random variate x, which is normally distributed
with a mean and standard deviation, is written in symbol form as N (µ, σ2
). It should be fully
understood that it is random errors alone which are treated by statistical processes.
1.10.2 Basic concept of errors
The basic concept of errors in the data captured by the surveyor may be likened to target shooting.
In the first instance, let us assume that a skilled marksman used a rifle with a bent sight, which
resulted in his shooting producing a scatter of shots as at A in Figure 1.23.
That the marksman is skilled (or reliable) is evidenced by the very small scatter, which illustrates
excellent precision. However, as the shots are far from the centre, caused by the bent sight (systematic
error), they are completely inaccurate. Such a situation can arise in practice when a piece of EDM
equipment produces a set of measurements all agreeing to within a few millimetres (high precision)
but, due to an operating fault and lack of calibration, the measurements are all incorrect by several
metres (low accuracy). If the bent sight is now corrected, i.e. systematic errors are minimized, the
result is a scatter of shots as at B. In this case, the shots are clustered near the centre of the target
and thus high precision, due to the small scatter, can be related directly to accuracy. The scatter is,
of course, due to the unavoidable random errors.
If the target was now placed face down, the surveyors’ task would be to locate the most probable
position of the centre based on an analysis of the position of the shots at B. From this analogy
several important facts emerge, as follows.
(1) Scatter is an ‘indicator of precision’. The wider the scatter of a set of results about the mean,
the less reliable they will be compared with results having a small scatter.
(2) Precision must not be confused with accuracy; the former is a relative grouping without regard
to nearness to the truth, whilst the latter denotes absolute nearness to the truth.
B
A
Fig. 1.23

(3) Precision may be regarded as an index of accuracy only when all sources of error, other than
random errors, have been eliminated.
(4) Accuracy may be defined only by specifying the bounds between which the accidental error of
a measured quantity may lie. The reason for defining accuracy thus is that the absolute error of
the quantity is generally not known. If it were, it could simply be applied to the measured
quantity to give its true value. The error bound is usually specified as symmetrical about zero.
Thus the accuracy of measured quantity x is x ± εx where εx is greater than or equal to the true
but unknown error of x.
(5) Position fixing by the surveyor, whether it be the coordinate position of points in a control
network, or the position of topographic detail, is simply an assessment of the most probable
position and, as such, requires a statistical evaluation of its reliability.
1.10.3 Further definitions
(1) The true value of a measurement can never be found, even though such a value exists. This is
evident when observing an angle with a one-second theodolite; no matter how many times the
angle is read, a slightly different value will always be obtained.
(2) True error (εx) similarly can never be found, for it consists of the true value (X) minus the
observed value (x), i.e.
X – x = εx
(3) Relative error is a measure of the error in relation to the size of the measurement. For instance,
a distance of 10 m may be measured with an error of ±1 mm, whilst a distance of 100 m may
also be measured to an accuracy of ±1 mm. Although the error is the same in both cases, the
second measurement may clearly be regarded as more accurate. To allow for this, the term
relative error (Rx) may be used, where
Rx = εx/x
Thus, in the first case x = 10 m, εx = ± 1 mm, and therefore Rx = 1/10 000; in the second case,
Rx = 1/100 000, clearly illustrating the distinction. Multiplying the relative error by 100 gives
the percentage error. ‘Relative error’ is an extremely useful definition, and is commonly used
in expressing the accuracy of linear measurement. For example, the relative closing error of a
traverse is usually expressed in this way. The definition is clearly not applicable to expressing
the accuracy to which an angle is measured, however.
(4) Most probable value (MPV) is the closest approximation to the true value that can be achieved
from a set of data. This value is generally taken as the arithmetic mean of a set, ignoring at this
stage the frequency or weight of the data. For instance, if A is the arithmetic mean, X the true
value, and εn the errors of a set of n measurements, then
A X
n
n
= –
[ ]ε
where [εn] is the sum of the errors. As the errors are equally as likely to be positive or negative,
then for a finite number of observations [εn]/n will be very small and A ≈ X. For an infinite
number of measurements, it could be argued that A = X. (N.B. The square bracket is Gaussian
notation for ‘sum of’.)
(5) Residual is the closest approximation to the true error and is the difference between the MPV
of a set, i.e. the arithmetic mean, and the observed values. Using the same argument as before,
it can be shown that for a finite number of measurements, the residual r is approximately equal
to the true error ε.

Table 1.1
Error Occurrence Probability
–0.10 1 1/121 = 0.0083
–0.09 2 2/121 = 0.0165
–0.08 3 3/121 = 0.0248
–0.07 4 4/121 = 0.0331
–0.06 5 5/121 = 0.0413
–0.05 6 6/121 = 0.0496
–0.04 7 7/121 = 0.0579
–0.03 8 8/121 = 0.0661
–0.02 9 9/121 = 0.0744
–0.01 10 10/121 = 0.0826
0 11 11/121 = 0.0909
0.01 10 10/121 = 0.0826
1.10.4 Probability
Consider a length of 29.42 m measured with a tape and correct to ± 0.05 m. The range of these
measurements would therefore be from 29.37 m to 29.47 m, giving 11 possibilities to 0.01 m for
the answer. If the next bay was measured in the same way, there would again be 11 possibilities.
Thus the correct value for the sum of the two bays would lie between 11 × 11 = 121 possibilities,
and the range of the sum would be 2 × ±0.05 m, i.e. between –0.10 m and +0.10 m. Now, the error
of –0.10 m can occur only once, i.e. when both bays have an error of –0.05 m; similarly with +0.10.
Consider an error of –0.08; this can occur in three ways: (–0.05 and –0.03), (–0.04 and –0.04) and
(–0.03 and –0.05). Applying this procedure through the whole range can produce Table 1.1, the
lower half of which is simply a repeat of the upper half. If the decimal probabilities are added
together they equal 1.0000. If the above results are plotted as error against probability the histogram
of Figure 1.24 is obtained, the errors being represented by rectangles. Then, in the limit, as the error
interval gets smaller, the histogram approximates to the superimposed curve. This curve is called
the normal probability curve. The area under it represents the probability that the error must lie
between ±0.10 m, and is thus equal to 1.0000 (certainty) as shown in Table 1.1.
More typical bell-shaped probability curves are shown in Figure 1.25; the tall thin curve indicates
small scatter and thus high precision, whilst the flatter curve represents large scatter and low
precision. Inspection of the curve reveals:
(1) Positive and negative errors are equal in size and frequency; they are equally probable.
(2) Small errors are more frequent than large; they are more probable.
(3) Very large errors seldom occur; they are less probable and may be mistakes or untreated
systematic errors.
The equation of the normal probability distribution curve is
y h h
= e
– –
1
2
2 2
π ε
where y = probability of an occurrence of an error ε, h = index of precision, and e = exponential
function.
As already illustrated, the area under the curve represents the limit of relative frequency, i.e.
probability, and is equal to unity. Thus tables of standard normal curve areas can be used to
calculate probabilities provided that the distribution is the standard normal distribution, i.e.

N(0, 12
). If the variable x is N(µ, σ2
), then it must be transformed to the standard normal distribution
using Z = (x – µ)/σ, where Z has a probability density function equal to (2 ) e
– – /2
1
2
2
π Z
when x = N(5, 22
) then Z = (x – 5)/2
When x = 9 then Z = 2
Thus the curve can be used to assess the probability or certainty that a variable x will fall between
certain values. For example, the probability that x will fall between 0.5 and 2.4 is represented by
area A on the normal curve (Figure 1.26(a)). This statement can be written as:
P(0.5 < x < 2.4) = area A
Now Area A = Area B – Area C (Figures 1.26(b) and (c))
where Area B represents P(x < 2.4)
0. 10
Probability(y)
0. 09
0. 08
0. 07
0. 06
0. 05
0. 04
0. 03
0. 02
0. 01
0.10 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.10
+ ∞∞–
Error (x)
Fig. 1.24
–1
–σs
+1
+σs
0
Fig. 1.25

and Area C represents P(x < 0.5)
i.e. P(0.5 < x < 2.4) = P(X < 2.4) – P(X < 0.5)
From the table of Standard Normal Curve Areas
When x = 2.4, Area = 0.9916
When x = 0.5, Area = 0.6915
∴ P(0.5 < x < 2.4) = 0.9916 – 0.6195 = 0.3001
That is, there is a 30.01% probability that x will lie between 0.5 and 2.4.
If verticals are drawn from the points of inflexion of the normal distribution curve (Figure 1.27)
they will cut that base at – σx and + σx, where σx is the standard deviation. The area shown indicates
the probability that x will lie between ±σx and equals 0.683 or 68.3%. This is a very important
statement.
Standard deviation (σx), if used to assess the precision of a set of data, implies that 68% of the
time, the arithmetic mean ( )x of that set should lie between (x x± σ ). Put another way, if the
Frequency
(a)
00.5 2.4
Values of measurement
Area A
(b)
Frequency
Area B
0 2.4
Frequency
(c)
Area C
00.5
Fig. 1.26

sample is normally distributed and contains only random variates, then 7 out of 10 should lie
between (x x± σ ). It is for this reason that two-sigma or three-sigma limits are preferred in
statistical analysis:
± =2 0.955 = 95.5% probabilityσ x
and ± =3 0.997 = 99.7% probabilityσ x
Thus using two-sigma, we can be 95% certain that a sample mean (x)will not differ from the
population mean µ by more than ± 2σ x . These are called ‘confidence limits’, where x is a point
estimate of µ and ( )x x2± σ is the interval estimate.
If a sample mean lies outside the limits of ± 2σ x we say that the difference between x and µ is
statistically significant at the 5% level. There is, therefore, reasonable evidence of a real difference
and the original null hypothesis (H x0 = ). µ should be rejected.
It may be necessary at this stage to more clearly define ‘population’and ‘sample’. The ‘population’
is the whole set of data about which we require information. The ‘sample’ is any set of data from
the population, the statistics of which can be used to describe the population.
1.11 INDICES OF PRECISION
It is important to be able to assess the precision of a set of observations, and several standards exist
for doing this. The most popular is standard deviation (σ), a numerical value indicating the amount
of variation about a central value.
In order to appreciate the concept upon which indices of precision devolve, one must consider a
measure which takes into account all the values in a set of data. Such a measure is the deviation
from the mean ( )x of each observed value (xi), i.e. ( – )x xi , and one obvious consideration would
be the mean of these values. However, in a normal distribution the sum of the deviations would be
zero; thus the ‘mean’ of the squares of the deviations may be used, and this is called the variance
(σ 2
).
σ ι
2
= 1
2
= ( – ) /Σi
n
x x n (1.1)
Theoretically σ is obtained from an infinite number of variates known as the population. In
practice, however, only a sample of variates is available and S is used as an unbiased estimator.
Account is taken of the small number of variates in the sample by using (n – 1)as the divisor, which
is referred to in statistics as the Bessel correction; hence, variance is
Frequency
68.3% of total area
–σx 0 +σx
Fig. 1.27

S x x n
i
n
i
2
= 1
2
= ( – ) / – 1Σ (1.2)
As the deviations are squared, the units in which variance is expressed will be the original units
squared. To obtain an index of precision in the same units as the original data, therefore, the square
root of the variance is used, and this is called standard deviation (S), thus
Standard deviation = S x x n
i
n
i= ( – ) – 1
= 1
2
1
2
±








Σ / (1.3)
Standard deviation is represented by the shaded area under the curve in Figure 1.27 and so
establishes the limits of the error bound within which 68.3% of the values of the set should lie, i.e.
seven out of a sample of ten.
Similarly, a measure of the precision of the mean ( )x of the set is obtained using the standard
error ( )Sx , thus
Standard error = S x x n n S nx
i
n
i= ( – ) – 1) = /
= 1
2
1
2
1
2±








Σ / ( (1.4)
Standard error therefore indicates the limits of the error bound within which the ‘true’ value of
the mean lies, with a 68.3% certainty of being correct.
It should be noted that S and Sx are entirely different parameters. The value of S will not alter
significantly with an increase in the number (n) of observations; the value of Sx , however, will
alter significantly as the number of observations increases. It is important therefore that to describe
measured data both values should be used.
Although the weighting of data has not yet been discussed, it is appropriate here to mention
several other indices of precision applicable to weighted (wi) data
Standard deviation (of weighted data)
= = ( – ) – 1
= 1
2
1
2
S w x x nw
i
n
i i±








Σ / (1.5)
Standard deviation of a single measure of weight wi
= = ( – ) ( – 1 = )
= 1
2
1
2 1
2S w x x w n S wwi
i
n
i i i w i±








Σ / /( (1.6)
Standard error (the weighted mean)
= = – ) ( ) – 1) =
= 1
2
= 1 = 1
1
2
1
2
S w x x w n S ww
i
n
i i
i
n
i w
i
n
i±













Σ Σ Σ( ( (1.7)
N.B. The conventional method of expressing sum of has been used for the various indices of precision,
as this is the format used in texts on statistics, and is therefore more easily recognizable. However,
for the majority of the expressions the neater Gaussian square bracket format has been used.
1.12 WEIGHT
Weights are expressed numerically and indicate the relative precision of quantities within a set.

The greater the weight, the greater the precision of the observation to which it relates. Thus an
observation with a weight of two may be regarded as twice as reliable as an observation with a
weight of one. Consider two mean measures of the same angle: A = 50° 50′ 50′′ of weight one, and
B = 50° 50′ 47′′ of weight two. This is equivalent to three observations, 50″, 47′′, 47′′, all of equal
weight, and having a mean value of
(50′′ + 47′′ + 47′′)/3 = 48′′
Therefore the mean value of the angle = 50° 50′ 48′′.
Inspection of this exercise shows it to be identical to multiplying each observation a by its
weight, w, and dividing by the sum of the weights [w], i.e.
Weighted mean = A
a w a w a w
w w w
aw
wm
n n
n
=
+ + . . +
+ + . . +
=
[ ]
[ ]
1 1 2 2
1 2
.
.
(1.8)
Weights can be allocated in a variety of ways, such as: (a) by personal judgement of the prevailing
conditions at the time of measurement; (b) by direct proportion to the number of measurements of
the quantity, i.e. w ∝ n; (c) by the use of variance and co-variance factors. This last method is
recommended and in the case of the variance factor is easily applied as follows. Equation (1.4)
shows
S S nx = /
1
2
That is, error is inversely proportional to the square root of the number of measures. However, as
w ∝ n, then
w Sx
2
1/∝
i.e. weight is proportional to the inverse of the variance.
1.13 REJECTION OF OUTLIERS
It is not unusual, when taking repeated measurements of the same quantity, to find at least one
which appears very different from the rest. Such a measurement is called an outlier, which the
observer intuitively feels should be rejected from the sample. However, intuition is hardly a scientific
argument for the rejection of data and a more statistically viable approach is required.
As already indicated, standard deviation S represents 68.3% of the area under the normal curve
and is therefore representative of 68.3% confidence limits. It follows from this that
±3.29S represents 99.9% confidence limits (0.999 probability)
Thus, any random variate xi, whose residual error ( – )x xi is greater than ±3.29 S, must lie in the
extreme tail ends of the normal curve and should therefore be ignored, i.e. rejected from the sample.
In practice, this has not proved a satisfactory rejection criterion due to the limited size of the
samples. Logan (Survey Review, No. 97, July 1955) has shown that the appropriate rejection criteria
are relative to sample size, as follows:

Sample size Rejection criteria
4 1.5 S
6 2.0 S
8 2.3 S
10 2.5 S
20 3.0 S
A similar approach to rejection is credited to Chauvenet. If a random variate xi, in a sample size
n has a deviation from the mean x greater than a 1/2n probability, it should be rejected. For
example, if n = 8, then 1/2n = 0.06 (94% or 0.94) and the probability of the deviate is 1.86S Thus,
an outlier whose residual error or deviation from the mean was greater than 1.86S. would be
rejected. This approach produces the following table:
Sample size Rejection criteria
4 1.53 S
6 1.73 S
8 1.86 S
10 1.96 S
20 2.24 S
It should be noted that successive rejection procedures should not be applied to the sample.
1.14 COMBINATION OF ERRORS
Much data in surveying is obtained indirectly from various combinations of observed data, for
instance the coordinates of a line are a function of its length and bearing. As each measurement
contains an error, it is necessary to consider the combined effect of these errors on the derived
quantity.
The general procedure is to differentiate with respect to each of the observed quantities in turn
and sum them to obtain their total effect. Thus if a = f (x, y, z, …), and each independent variable
changes by a small amount (an error) δx, δy, δz …., then a will change by a small amount equal to
δa, obtained from the following expression:
δ ∂
∂
δ ∂
∂
δ ∂
∂
δa a
x
x a
y
y a
z
z= + + + . . .⋅ ⋅ ⋅ (1.9)
in which ∂a/∂x is the partial derivative of a with respect to x, etc.
Consider now a set of measurements and let xi = δxi, yi = δyi, zi = δz, equals a set of residual errors
of the measured quantities and ai = δai:
a a
x
x a
y
y a
z
z1 1 1 1= + + +. . .∂
∂
∂
∂
∂
∂
⋅ ⋅ ⋅
a a
x
x a
y
y a
z
z2 2 2= + + +. . .∂
∂
∂
∂
∂
∂
⋅ ⋅ ⋅2
M M M M
a a
x
x a
y
y a
z
zn n n n= + + +. . .∂
∂
∂
∂
∂
∂
⋅ ⋅ ⋅

Now squaring both sides gives
a a
x
x a
x
a
y
x y a
y
y1
2
2
1 1
2
= + 2 + . . + . .∂
∂
∂
∂
∂
∂
∂
∂



 ⋅ 














1
2
1
2
. .
a a
x
x a
x
a
y
x y a
y2
2
2
2
2
2 2
2
2
2
= + 2 + . . y + . .∂
∂
∂
∂
∂
∂
∂
∂



 ⋅ 














. .
a a
x
x a
x
a
y
x y a
y
yn n n n n
2
2
2
2
2
= + 2 + . . + . .∂
∂
∂
∂
∂
∂
∂
∂



 ⋅ 














. .
In the above process many of the square and cross-multiplied terms have been omitted for simplicity.
Summing the results gives
[ ] . ] .a a
x
x a
x
a
y
xy a
y
y2 2 2
= [ ] + 2 [ ] + . . [ + . .∂
∂
∂
∂
∂
∂
∂
∂




















As the measured quantities may be considered independent and uncorrelated, the cross-products
tend to zero and may be ignored.
Now dividing throughout by (n – 1):
[ ] ]
.
a
n
a
x
x
n
a
y
y
n
a
z
z
n
2 2 2 2
]
– 1
=
[
– 1
+
[ ]
– 1
+
[
– 1
+ . .∂
∂
∂
∂
∂
∂









 



The sum of the residuals squared divided by (n – 1), is in effect the variance σ 2
, and therefore
σ ∂
∂
σ ∂
∂
σ ∂
∂
σa
2
x
2
y z
a
x
a
y
a
z
= + + + . .2 2








 



. (1.10)
which is the general equation for the variance of any function. This equation is very important and
is used extensively in surveying for error analysis, as illustrated in the following examples.
1.14.1 Errors affecting addition or subtraction
Consider a quantity A(f ) = a + b where a and b are affected by standard errors σa and σb, then
σ
∂
∂
σ
∂
∂
σ σ σ σ σ σA a b a b A a b
a b
a
a b
b
2
2
2 2 2 2
=
( + )
+
( + )
= + = ( + )
1
2











∴ ±
2
(1.11)
As subtraction is simply addition with the signs changed, the above holds for the error in a
difference:
If σa = σb = σ, then σ σA n= ( )
1
2± (1.12)
Equation (1.12) should not be confused with equation (1.4) which refers to the mean, not the sum
as above.
Worked examples
Example 1.1 If three angles of a triangle each have a standard error of ±2′′, what is the total error
(σT) in the triangle?
σ T = (2 + 2 + 2 = 2(3) = 3.52 2 2
1
2
1
2± ± ± ′′)

Example 1.2 In measuring a round of angles at a station, the third angle c closing the horizon is
obtained by subtracting the two measured angles a and b from 360°. If angle a has a standard error
of ±2″ and angle b a standard error of ±3″, what is the standard error of angle c?
c ± σc = 360° – (a ± σa) – (b ± σb)
= 360° – (a ± 2″) – (b ± 3″)
since c = 360° – a – b
then ±σc = ±σa ±σb = ±2″ ±3″
and σ c = (2 + 3 ) = 3.62 2
1
± ± ′′2
Example 1.3 The standard error of a mean angle derived from four measurements is ±3″; how
many measurements would be required, using the same equipment, to halve this error?
From equation (1.4) σ
σ
σm
s
s
n
= = 3 4 = 61
2
1
2± ∴ × ± ′′
i.e. the instrument used had a standard error of ±6″ for a single observation; thus for σm = ±1.5″,
when σs = ±6″
n =
6
1.5
= 16
2




Example 1.4 If the standard error of a single triangle in a triangulation scheme is ±6.0″, what is the
permissible standard error per angle?
From equation (1.12) σ σT p n= ( )
1
2
where σT is the triangular error, σp the error per angle, and n the number of angles.
∴ σ
σ
p
T
(n
=
)
=
6.0
(3)
= 3.51
2
1
2
± ′′
± ′′
1.14.2 Errors affecting a product
Consider A(f) = (a × b × c) where a, b and c are affected by standard errors. The variance
σ
∂
∂
σ
∂
∂
σ
∂
∂
σA a b c
abc
a
abc
b
abc
c
2
2 2 2
=
( )
+
( )
+
( )

















= (bcσa)2
+ (acσb)2
+ (abσc)2
∴ σ
σ σ σ
A
a b c
abc
a b c
= + +
2
1
2
± 

















2 2
(1.13a)
The terms in brackets may be regarded as the relative errors Ra, Rb, Rc giving
σ A a b cabc R R R= ( + + )2 2
1
2± 2 (1.13b)

1.14.3 Errors affecting a quotient
Consider A(f) = a/b, then the variance
σ
∂
∂
σ
∂
∂
σ
σ σ
A a b
a bab
a
ab
b b
a
b
2
–1 –1 2 2
2
2
=
(
+
( )
= +
)



















2
∴ σ
σ σ
A
a ba
b a b
= +
2 2
1
2
± 













(1.14a)
= +2 2
1
2± a
b
R Ra b( ) (1.14b)
1.14.4 Errors affecting powers and roots
The case for the power of a number must not be confused with multiplication, since a3
= a × a × a,
with each term being exactly the same.
Thus if A(f) = an
, then the variance
σ ∂
∂
σ σA
n
a
n
a
a
a
na2 –1
= = (



2
2
) ∴ σA = ±(nan–1
σa) (1.15a)
Alternatively R
a
na
a
n
a
nRA
A
n
n
a
n
a
a= = = =
– 1
σ σ σ
(1.15b)
Similarly for roots, if the function is A(f) = a1/n
, then the variance
σ ∂
∂
σ σ σA
n
a
n
a
n
a
a
a n
a
n
a a2
1/ 2
1/ –1 1/ –1
2
= =
1
=
1











2
= =
1/ 1/
a
n a
a
n a
n
a
A
n
aσ
σ
σ



∴ ± 



2
(1.16)
The same approach is adopted to general forms which are combinations of the above.
Worked examples
Example 1.5 The same angle was measured by two different observers using the same instrument,
as follows:
Observer A Observer B
° ′ ″ ° ′ ″
86 34 10 86 34 05
33 50 34 00
33 40 33 55
34 00 33 50
33 50 34 00
34 10 33 55
34 00 34 15
34 20 33 44

Engineering surveying, 5...ition w. schofield

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